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These are great questions!
1) You will see multiple modes excited at the output of the fiber, because of mode coupling at the input of the multimode fiber. You can check the Help on the Parabolic-Index Multimode Fiber to see how this coupling is modeled. The input to the fiber is a single mode, but obviously it is not exactly the same as the fundamental mode of the fiber. This mismatch between the modes can cause higher order modes to be excited along with the fundamental fiber mode. This is an interesting topic and if you want to learn more I would suggest looking at this link:
2) The power of higher order modes will usually be less than the fundamental mode, but this really depends on how your input mode(s) compare to the modes supported by a fiber. To determine the power you can use the Mode Selector component and a power meter. I already made a project file, that could be easily adapted for this application, in a different forum post:
3) The spot size at the output is simply due to the fact that the fundamental mode of the fiber has a larger spot size than the mode supported by the Spatial CW Laser. Since these modes are propagating in a guided medium, theoretically there would be no beam waist expanding.
4) The best way to try and insure single mode behavior in a multimode fiber, is to match the incident laser mode to the fundamental mode of the fiber. Depending on the laser you are using in the lab, you could use a lens to focus the light into a smaller area or expand the beam into a larger area. OptiSystem has components to model this behavior. Of course, in a real optical fiber, bends and imperfections will, over longer distances, couple power away from the fundamental mode and into higher order modes.
Hope this helps!